All Questions
Tagged with pr.probability stochastic-differential-equations
237 questions
3
votes
1
answer
191
views
Stopping time property
Hi, I am reading a textbook about SDE, and am very confused about the transition
$$X_T 1_{T\lt t} + E\{X_T 1_{T\geq t} | F_{t\wedge T}\}$$
$$= X_T 1_{T\lt t} + E\{X_T | F_t\} 1_{T\geq t}$$
I ...
- 39
4
votes
1
answer
302
views
Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...
- 1,675
2
votes
1
answer
280
views
Walker whose Velocity is a Brownian Bridge
Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...
- 1,444
4
votes
1
answer
322
views
Asymptotic form of pdf of Escape Time of arithmetic fBm
I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
- 291
2
votes
1
answer
534
views
Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
- 51
2
votes
0
answers
104
views
Stochastic stability of "open" continuous-time stochastic systems: reference request
I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...
- 1,675
4
votes
1
answer
509
views
Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
3
votes
1
answer
2k
views
Solution of multivariate Geometric Brownian Motion?
It is known how to solve the SDE $dX=X\,dW$ to get a closed form expression of $X(t)=\exp(W_t-\frac{t}{2})$. The question is, is there also a way to solve
\begin{equation} \begin{cases}
dX=X \, dW_1+...
- 91
4
votes
1
answer
386
views
Existence of normal number except random numbers
For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...
- 3,094
1
vote
0
answers
90
views
Onsager-Machlup Function of a Killed Diffusion Process
Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
- 213
1
vote
0
answers
340
views
Construction of the quadratic variation for Hilbert space valued local martingales
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167
4
votes
1
answer
320
views
Uniqueness of a SDE with positivity constraint
We start by fixing some notation.
If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
- 779
3
votes
1
answer
751
views
Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined
Remark: I've asked this question on MSE as well.
Let
$T>0$
$I:=[0,T]$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
- 167
3
votes
2
answers
380
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
- 719
3
votes
1
answer
83
views
Filtering Mixed Discrete and Continous
Suppose I have signal process $\lambda_t$ following the dynamics
\begin{equation}
\begin{aligned}
\zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\
\xi_t&=\mu^{\xi}(...
- 5,405
1
vote
0
answers
249
views
Ito's formula for jump diffusions
Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...
- 11
3
votes
1
answer
110
views
Sequence of diffusions
Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?
- 5,405
2
votes
1
answer
212
views
Modified square root process
I am dealing with the following stochastic differential equation (SDE)
$ \begin{cases} dS_t &= \mu S_t dt + \sigma_1 S_tdW^1_t\\dG_t &= kS_t(\alpha - G_t)dt + \sigma_2\sqrt{G_tS_t}dW^2_t
\end{...
- 21
1
vote
0
answers
57
views
Matching Numbers in Ito McKean
Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \...
- 163
2
votes
1
answer
387
views
Weak convergence of sum of log normal random variables
Let $S_t$ be the Geometric Brownian Motion, we know that
$$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$
and the distribution of $S_t$ is known explicitly. Please see the ...
- 323
1
vote
1
answer
924
views
Solutions to linear SDE with many noise sources
It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...
- 749
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
3
votes
2
answers
1k
views
Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t)
\...
- 203
3
votes
0
answers
170
views
Feynman-Kac formula for *general* Sturm-Liouville operator
One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...
- 891
0
votes
3
answers
639
views
Non-smooth Ito lemma for semi-martingales
Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any ...
- 5,405
0
votes
0
answers
57
views
Parametric distribution where the parameter follows a diffusion process
I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...
- 1,902
1
vote
0
answers
260
views
Transforming reaction-diffusion equations to random walk processes
I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...
- 53
3
votes
2
answers
923
views
On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure
Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
- 31
0
votes
0
answers
153
views
Embedding a martingale by SDE
Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE:
$$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\...
- 1,835
3
votes
0
answers
78
views
Perscribed/Inverting Conditional Expectation
I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
- 5,405
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
1
vote
0
answers
29
views
Usually trivial Excursion-type process
How Can i construct a stochastic process $X_t$ which has the property that:
$X_t \in [0,1]$ for all $t \in [0,T]$ and
$m(\{t \in [0,T] : X_t>0 \})\leq \delta$, for some pre-chosen $\delta \in [0,T]...
2
votes
1
answer
148
views
Joint distribution of integrals of diffusion and driving noise
Consider a generic diffusion of the form
$$dX_t=f(t,X_t)dt+dB_t,$$
where $f$ is some nice function and $B_t$ is a standard Brownian motion.
The marginal distributions of the integrals
$$I:=\int_0^...
- 891
2
votes
1
answer
139
views
Stochastic inverse
Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying
$$
\int_0^Tf(H_t,...
- 5,405
4
votes
1
answer
463
views
Variance and expectation of timed-change squared Bessel process
Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
- 323
2
votes
0
answers
52
views
Minimizer of a class of SDEs
Setup
Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...
- 5,405
6
votes
2
answers
748
views
Does there exist a stochastic time derivative?
The Setup
Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE
$$
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
$$
and $f$ is a smooth function.
My Question
Is there a ...
- 5,405
3
votes
1
answer
2k
views
On a reflecting Brownian motion and its boundary local time
I have a question about a reflecting Brownian motion and its boundary local time.
Bass and Hsu studied the existence of Reflecting Brownian motion and boundary local time on a bounded Lipschitz ...
- 721
1
vote
1
answer
216
views
Sufficient condition of continuity of the expected stopping time
Let $\sigma \in C(\mathbb R)$, and $X$ be a solution of
\begin{equation}\label{eq:1}
X_{t} = x + t + \int_{0}^{t} \sigma(X_{s}) dB_{s}
\end{equation}
where $B$ is 1-d Brownian motion under filtered ...
- 1,399
2
votes
3
answers
471
views
Existence of solution to SDE with perscribed initial and terminal conditions
The SDEs \begin{equation}
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
\end{equation} with prescribed initial conditions are well studied. My question came up in my research and I have not found much on ...
- 5,405
1
vote
1
answer
913
views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
- 247
2
votes
0
answers
221
views
Boundary behavior for Ito diffusions
The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
3
votes
1
answer
159
views
Differentiability of a simple value function driven by a diffusion
Consider a diffusion given by,
$d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$
$X_0 = x$.
Suppose the functions $\mu$ and $\sigma$ are as follows -
$f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
- 553
2
votes
0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 155
2
votes
1
answer
219
views
The (infinite) invariant measure of an SPDE
Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type:
\begin{equation}\left\{
\begin{aligned}
&\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
- 199
4
votes
1
answer
405
views
Limit of first passage time
I have a conjecture that seems rather obvious but the proof seems elusive.
Consider a diffusion given by,
$dX_t = \mu(X_t) dt + \sigma(X_t) db_t$
where $b_t$ is a standard Brownian motion.
$\mu,\...
- 553
1
vote
1
answer
375
views
Computing transition operators for Markov processes
Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$
(or given by ...
- 138
5
votes
1
answer
828
views
Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
- 503
1
vote
1
answer
762
views
Feller property for Ito diffusion with Lipschitz coefficients
Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded ...
- 503
2
votes
0
answers
96
views
Smoothness of Value function for SDE with discontinuous coefficients
Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...
- 21